Hey everyone.
I am trying to define a continuous branch of log z in the
complex plane slit along the positive axis: C\{[0,i*infinity]}
I understand that we need to make a branch cut because log z will not be continuous on all of C. When the cut is the negative real axis I think I understand. And I also realize that the the cut can be any ray by my intuition. But I don't understand how to put it down as a function in this case.
I see, it is more straightforward than I thought. Thanks a lot.
How about this one:
I have to show that
|cosh z|^2 = cos^2(y)+sinh^2(x)
I have tried writing out cosh z in terms of e^x and e^y so that I can take the modulus the old fashioned way (x^2+y^2)^(1/2).
(from the e-definition of cosh z and using that e^z=e^xcos(y)+ie^xsin(y))
but the resulting mess is overwhelming and I can't seem to split up the real and imaginary parts neatly.